BIEBERBACH GROUPS WITH FINITE POINT GROUPS NOR ASHIQIN BINTI MOHD IDRUS UNIVERSITI TEKNOLOGI MALAYSIA

Transcription

1 BIEBERBACH GROUPS WITH FINITE POINT GROUPS NOR ASHIQIN BINTI MOHD IDRUS UNIVERSITI TEKNOLOGI MALAYSIA

2 BIEBERBACH GROUPS WITH FINITE POINT GROUPS NOR ASHIQIN BINTI MOHD IDRUS A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy (Mathematics) Faculty of Science Universiti Teknologi Malaysia APRIL 2011

3 iii To my beloved husband, children and mother

4 iv ACKNOWLEDGEMENT First of all, I would like to thank Allah s.w.t for giving me the strength and blessings throughout my studies. It was His will that this thesis was completed. I would like to express my deepest gratitude to my supervisor, Assoc. Prof. Dr. Nor Haniza Sarmin for her continuous guidance, support and patience during this research. She is the best supervisor that I would recommend to friends. My heartfelt gratitude and thanks also go to my external supervisor Prof. Dr. Robert F. Morse in appreciation of the significant contribution, encouragement and guidance which he provided throughout the completion of this thesis. I am grateful for his valuable supervison through video conferences and while I was in University of Evansville for a three-week visit ( 2-23 February 2008) followed by his two-week visit to UPSI as a Visiting Professor from 5 June to 17 June I would also like to thank UPSI for approving my sabbatical leave from 1 January to 30 September 2010 (9 months) in which I had the opportunity to spend three months ( 1 January - 20 February 2010 and 26 August - 30 September 2010) with Assoc. Prof Dr Nor Haniza at UTM and six months (24 February - 22 August 2010) with Prof. Dr. Morse at University of Evansville. Acknowledgement also to Research Management Center, UPSI and to the Ministry of Higher Education (MOHE) for funding related researches through the University fund Vote No and The Fundamental Research Grant Scheme (FRGS) Vote No respectively. My special thanks to my husband, children and my family for their continuous support and patience, great loves and for always being there with me in undergo the hard times together. Last but not least, thanks to my three dearest friends Rohaidah, Azhana and Nor Muhainiah who have helped me in many ways in the completion of this thesis.

5 v ABSTRACT A Bieberbach group is a torsion free crystallographic group. It is an extension of a lattice group, which is a maximal normal free abelian group of finite rank, by a finite point group. The main objective of this research is to compute the nonabelian tensor square of Bieberbach groups with a finite nonabelian point group, in particular the dihedral group of order eight. Bieberbach groups in the Crystallographic AlgoRithms And Tables (CARAT) homepage were first explored and examples of the nonabelian tensor square of the groups were then computed by using the Groups, Algrorithms, Programming (GAP) software system. The exploration of the groups and the examples computed led to the exact characterization of the Bieberbach groups with trivial center. The centerless Bieberbach groups are interesting since they do not arise in the general construction of a Bieberbach group for a given point group. This construction has been shown to depend on the presentation of the point group. In addition, the experimental data of the computation of the nonabelian tensor square gives no insight into the structure of the tensor square such as its generators and relations. With the method developed for polycyclic groups, the nonabelian tensor square of one of the centerless Bieberbach groups with dihedral point group of order eight were manually computed. It has been demonstrated that the use of GAP helps to simplify the manual calculation. Furthermore, the computation of some homological functors of all 73 centerless Bieberbach groups with dihedral point group of order eight and of dimension at most six were explored. Lastly, some homological functors for Bieberbach groups with some other nonabelian point groups were also computed with the help of GAP.

6 vi ABSTRAK Kumpulan Bieberbach adalah kumpulan kristolografi yang bebas kilasan. Ia adalah perluasan kepada kumpulan kekisi iaitu kumpulan abelan bebas yang normal dan maksimal, melalui kumpulan titik terhingga. Objektif utama kajian ini adalah untuk mengira kuasa dua tensor tak abelan bagi kumpulan Bieberbach dengan kumpulan titik tak abelan berperingkat terhingga, perincian kepada kumpulan titik dwihedron berperingkat lapan. Kajian ini dimulakan dengan meneroka kumpulan tersebut dalam laman Crystallographic AlgoRithms And Tables (CARAT) dan seterusnya pengiraan contoh-contoh kuasa dua tensor tak abelan kumpulan tersebut dibuat dengan menggunakan sistem perisian Groups, Algrorithms and Programming (GAP). Hasil penerokaan dan contohcontoh pengiraan menghala ke arah pencirian kumpulan Bieberbach dengan pusat remeh. Kumpulan Bieberbach tidak berpusat adalah menarik kerana ia tidak wujud dalam pembinaan umum kumpulan Bieberbach untuk kumpulan titik yang diberi. Pembinaan ini ditunjukkan bergantung kepada persembahan kumpulan titik tersebut. Tambahan lagi, data eksperimental bagi pengiraan kuasa dua tensor tak abelan tidak memberi maklumat yang mendalam tentang struktur kumpulan tersebut seperti penjana dan perhubungannya. Dengan menggunakan kaedah kumpulan polikitaran, kuasa dua tensor tak abelan bagi salah satu kumpulan Bieberbach yang tidak berpusat dengan kumpulan titik dwihedron berperingkat lapan telah dilakukan secara manual. Penggunaan GAP ditunjukkan dalam membantu memudahkan pengiraan secara manual tersebut. Selanjutnya, pengiraan beberapa fungtor berhomologi bagi kesemua 73 kumpulan Bieberbach dengan kumpulan titik dwihedron berperingkat lapan dan berdimensi enam atau kurang dikaji. Akhir sekali, beberapa fungtor berhomologi untuk kumpulan Bieberbach dengan kumpulan titik tak abelan yang lain juga dikira dengan bantuan GAP.

7 vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS ii iii iv v vi vii xi xii xiii 1 INTRODUCTION Introduction Research Background Problem Statement Research Objectives Scope of Thesis Significance of Findings Thesis Outline 5 2 LITERATURE REVIEW Introduction 7

8 viii 2.2 Crystallographic Groups and Bieberbach Groups Characterization of Bieberbach Groups The Nonabelian Tensor Squares of Groups Computing the Nonabelian Tensor Squares of Groups GAP and CARAT Conclusion 15 3 PRELIMINARY RESULTS Introduction Definitions and Notations Some Basic Results Some Structural Results Conclusion 29 4 CONSTRUCTING A BIEBERBACH GROUP WITH AN ARBITRARY FINITE POINT GROUP Introduction The Construction of a Bieberbach Group With An Arbitrary Finite Point Group Examples of The Constructions of The Bieberbach Groups With Certain Point Groups The Construction of The Bieberbach Group With Point Group C The Construction of The Bieberbach Group With Point Group D Conclusion 44 5 THE STRUCTURE OF THE ABELIANIZATION OF A BIEBERBACH GROUP Introduction 45

9 ix 5.2 The Abelianization of a Bieberbach Group With Any Finite Point Group Conclusion 51 6 THE NONABELIAN TENSOR SQUARE OF CENTERLESS BIEBERBACH GROUPS WITH DIHEDRAL POINT GROUP: THEORY AND CALCULATION Introduction Centerless Bieberbach Groups With Dihedral Point Groups of Order Eight, D 8 and Their Nonabelian Tensor Squares: Theory The Nonabelian Tensor Squares of Centerless Bieberbach Groups With Dihedral Point Groups of Order Eight, D 8 : Computation Conclusion 86 7 EXPERIMENTAL RESULTS FROM GAP Introduction Homological Functors of All Centerless Bieberbach Groups With Dihedral Point Group of Order Eight Torsion Subgroups of The Nonabelian Tensor Squares of Bieberbach Groups With Nonabelian Point Groups Conclusion 97 8 CONCLUSION Summary of The Research Suggestions for Future Research 99

10 x REFERENCES 101 Appendices A - C

11 xi LIST OF TABLES TABLE NO. TITLE PAGE 7.1 Homological functors for 73 centerless Bieberbach groups Bieberbach groups with nonabelian point groups 97

12 xii LIST OF FIGURES FIGURE NO. TITLE PAGE 2.1 The commutative diagram of homological functors 12

13 xiii LIST OF SYMBOLS 1 - Identity element Ker(f) - Kernel of mapping f A B - A is a subgroup of B a - subgroup generated by an element a A B - A is a normal subgroup of B G/H - the quotient group G by H where H G A B - direct product of A and B x y - x conjugate by y, y 1 xy [a, b] - commutator of a and b, a 1 b 1 ab H = G - H is isomorphic to G G G - nonabelian exterior square of G H 2 (G) - Schur Multiplier of G C n - Cyclic group of order n X R - Group presented by generators X and relators R H K - Tensor product of H and K G = [G, G] - Derived subgroup of G G/G - The abelianization of G Z(G) - Center of G x G - x is an element of G x / G - x is not an element of G F n - free group of rank n Fab n - free abelian group of rank n h(g) - Hirsch length of G

14 xiv LIST OF APPENDICES APPENDIX TITLE PAGE A B The computations of homological functors for 73 centerless Bieberbach groups 104 The computations of the torsion subgroups of the nonabelian tensor squares of Bieberbach groups with point group S 3, D 8, A 4 and Q C Publication/Presentation in seminars/conferences 115

15 CHAPTER 1 INTRODUCTION 1.1 Introduction A free abelian group G is the extension 1 A G 1 1 where the point group is finite (trivial). A crystallographic group G is a generalization of free abelian groups since it satisfies the short exact sequence 1 L G P 1 where P is a point group that is a finite group acting faithfully on a maximal normal free abelian subgroup L of G which is of finite rank. The subgroup L is called a lattice group. It follows that L is a Fitting subgroup of G and its rank or Hirsch length is referred to as the dimension of G. Crystallographic groups arise as discrete, irreducible subgroups of the group of isometries of the n dimensional Euclidean space. They are used to study the structures and characteristics of crystals. One of the crystallographic point groups that the researchers had an interest is Bieberbach groups. Bieberbach groups are torsion-free crystallographic groups. The nonabelian tensor square G G of a group G is a special case of the nonabelian tensor product G H for two arbitrary group G and H that was

16 2 introduced by Brown and Loday [1] extending the idea of Whitehead [2]. The nonabelian tensor square G G is a group generated by the symbols g h for g, h G, subject to relations gh k = (g h k h )(h k) and g hk = (g k)(g k h k ) for all g, h, k G where g h = h 1 gh. It is defined by a presentation on G 2 generators and 2 G 3 relations. This presentation does not reflect the group structure very well. The structure of G G has been first studied by Brown, Johnson and Robertson [3] where they investigated the group structures in terms of central extensions. Their focus is to compute the nonabelian tensor square of groups by finding a simplified presentation for the nonabelian tensor square from the presentation given by the definition. 1.2 Research Background Bieberbach groups are torsion-free crystallographic groups. They appear as fundamental groups of compact, connected, flat Riemannian manifolds and have many interesting algebraic properties [4, 5]. Any new properties or results concerning crystallographic groups, particularly Bieberbach groups might lead to new exploration of the groups by not only mathematicians but by physicists and chemists too. New properties and results of the groups can be obtained by, not limited to, exploring the groups and by computing their nonabelian tensor squares. Eventhough the work on computing the nonabelian tensor squares of groups have started a long time ago, but the work on computing the nonabelian tensor squares of Bieberbach groups has just started by Rohaidah [6] in She computed the nonabelian tensor squares of Bieberbach groups with cyclic point groups. One of the main goals of this research is to compute the nonabelian tensor squares of Bieberbach groups with a nonabelian point group, particularly with a dihedral point group of order eight. The exploration of the groups in the

17 3 Crystallographic AlgoRithms And Tables (CARAT) [7] and the computation of the nonabelian tensor squares of the groups with the Groups, Algrorithms, and Programming (GAP) software [8] lead us first to characterize exactly the general Bieberbach group with trivial center. The centerless Bieberbach groups are of interest since they do not arise in the general construction of a Bieberbach group for any given finite point group. It will be shown that the construction depends on the presentation of the point group. 1.3 Problem Statement To explore the properties of Bieberbach groups with any finite point group and to calculate the nonabelian tensor squares of Bieberbach groups with nonabelian point group; in particular, the dihedral point group of order eight. 1.4 Research Objectives The objectives of this research are (i) to construct a Bieberbach group with an arbitrary finite point group (ii) to give examples of the construction of Bieberbach groups in (i) with a cyclic point group of order two and with a dihedral point group of order eight (iii) to characterize Bieberbach groups with trivial center as exactly those with finite abelianization (iv) to determine the properties of centerless Bieberbach groups with a dihedral point group of order eight and their nonabelian tensor squares

18 4 (v) to compute the nonabelian tensor squares of a centerless Bieberbach group of dimension four with dihedral point group of order eight and whose derived length of its nonabelian tensor squares is two and (vi) to compute some homological functors of Bieberbach groups with dihedral point group of order eight and with other nonabelian point groups with GAP. 1.5 Scope of Thesis In this thesis, only n dimension Bieberbach groups with any point group are considered. Properties of only centerless Bieberbach groups with dihedral point group of order eight and their nonabelian tensor squares are obtained. In computing the nonabelian tensor squares of a group, a group is limited to a centerless Bieberbach group of dimension four with a dihedral point group of order eight in which the derived length of its nonabelian tensor squares is two. 1.6 Significance of Findings The major contribution of this thesis will be new theoretical results on constructing and characterizing the Bieberbach groups with any finite point groups. The properties of the centerless Bieberbach groups of dimension four with dihedral point group of order eight and computing their nonabelian tensor squares will contribute as a foundation in determining the generalization of their nonabelian tensor squares and the nonabelian tensor squares of Bieberbach groups with other nonabelian finite point groups. Thus this thesis contributes to new findings in the field of theoretical and computational group theory.

19 5 1.7 Thesis Outline There are eight chapters in this thesis. Chapter 1 presents the introduction of the thesis. This chapter discusses research background, problem statement, research objectives, research scope and the significance of the thesis. In Chapter 2, the studies of crystallographic groups especially Bieberbach groups are presented. Some characterizations of Bieberbach groups given by several researchers are discussed. The background of the nonabelian tensor squares of groups are overviewed and the methods of computing the nonabelian tensor squares of various groups by several researchers are compared. The method of computing the nonabelian tensor squares initiated by Rocco [9] followed by Ellis and Leonard [10] and extended by Blyth and Morse [11] is presented briefly in this chapter. The background and the application of the software system for computational group theory, GAP, and a computer package consists of a library of Bieberbach groups, CARAT, are also presented in this chapter. Chapter 3 is a chapter of preliminary results. It presents some related and important definitions in group theory that are used throughout the thesis. Some basic results on free groups are presented in this chapter. Methods chosen in computing the nonabelian tensor squares of polycyclic groups and related results that are used in the thesis are elaborated deeply in one of the section in this chapter. A list of commutator calculus is also given here for easy reference. Chapter 4 discusses the main result on constructing a Bieberbach group with an arbitrary point group. The proof of the existence of a Bieberbach group with any finite point group are presented in this chapter. With the construction discussed, examples of Bieberbach groups with a cyclic point group of order two and a dihedral point group of order eight are given here. Chapter 5 presents a new characterization of any Bieberbach group with finite point group. The characterization is based on the structure of the abelianization of a centerless Bieberbach group.

20 6 Chapter 6 discusses the theory and the calculation of the nonabelian tensor squares of centerless Bieberbach groups with a dihedral point group of order eight. The theory gives us the properties of the groups and its nonabelian tensor squares. In this chapter, with the method chosen, the computation of the nonabelian tensor squares of a centerless Bieberbach group of dimension four with a dihedral point group of order eight in which the derived length of its nonabelian tensor squares is two is presented. The polycyclic presentation of the group is shown to be consistent so that GAP can be used to assist the hand computation. The nonabelian tensor square of the group is presented by a simple presentation of its generators and relations. The exploration of some homological functors of all centerless Bieberbach groups with a dihedral point group of order eight with GAP is discussed in Chapter 7. In this chapter the exploration is not limited to the centerless Bieberbach groups but also to Bieberbach groups with other nonabelian point groups. Examples of GAP codes for the purposes are presented in this chapter. Few results regarding some of the homological functors of the groups are also presented. Lastly, the thesis is summarized in Chapter 8. Some suggestions for future research on the nonabelian tensor square of the Bieberbach groups and other homological functors of the groups are given in this chapter.

21 101 REFERENCES 1. Brown, R. and Loday, J-L. Van Kampens Theorems for Diagrams of Spaces. Topology, (3): Whitehead, J. H. C. A Certain Exact Sequence. Annals of Mathematics (2), : Brown, R., Johnson, D. L. and Robertson, E. F. Some Computations of Nonabelian Tensor Products of Groups. Journal of Algebra, : Auslander, L. and Kuranishi, M. On The Holonomy Group of Locally Euclidean Spaces. Annals Of Mathematics (3), : Hiller, H. Crystallography and Cohomology of Groups. The American Mathematical Monthly, : Rohaidah Masri. The Nonabelian Tensor Squares of Certain Bieberbach Groups With Cyclic Point Groups. Ph.D. Thesis. Universiti Teknologi Malaysia; Torsion Free Space Groups. ( bieberbach.html). 8. The GAP Group. GAP-Groups, Algorithms, and Programming. Version : ( 9. Rocco, N. R. On a Construction Related to the Nonabelian Tensor Square of a Group. Bol. Soc. Brasil. Mat. (N.S.), (1): Ellis, G. and Leonard, F. Computing Schur Multipliers and Tensor Products of Finite Groups. Proc. Roy. Irish Acad. Sect. A., (2):

22 Blyth, R. D. and Morse, R. F. Computing the Nonabelian Tensor Squares of Polycyclic Groups. Journal of Algebra, : Farkas, D.R. Crystallographic Groups and Their Mathematics. Journal of Mathematics, 1981, 11: Auslander, M. and Lyndon, R.C. Commutator Subgroups of Free Groups. American Journal of Mathematics, : Szczepanski, A. Outer Automorphism Groups of Bieberbach Groups. Bull. Belg. Math. Soc. (3), : Malfait, W. and Szczepanski, A. The Structure of The (Outer) Automorphism Groups of a Bieberbach Group. Compositio Mathematica, : Putryez, B. Commutator Subgroup of Hantzsche-Wendt Groups. Journal of Group Theory, : Bacon, M. R. and Kappe, L. C. The Nonabelian Tensor Square of a 2- Generator p-group of Class Two. Arch. Math, : Beuerle, J. R. and Kappe, L. C. Infinite Metacyclic Groups and Their Nonabelian Tensor Squares. Proc. Edinburgh Math. Soc. (2), (3): Bacon, M. R. and Kappe, L. C. and Morse, R. F. On the Nonabelian Tensor Square of a 2-Engel Group. Archiv de Mathematik, : Blyth, R. D., Morse R. F. and Redden, J. L. On Computing the Nonabelian Tensor Squares of the Free 2-Engel Groups. Proc. Edinb. Math. Soc. (2), (2): Sarmin, N. H. Infinite Two-generator Groups of Class Two and Their Nonabelian Tensor Squares. IJMMS, (10): Bacon, M. R. On the Nonabelian Tensor Square of a Nilpotent Group of Class Two. Glasgow Mathematical Journal, (3):

23 Kappe, L. C., Sarmin, N. H. and Visscher, M. P. Two-generator Two-groups of Class Two and Their Nonabelian Tensor Squares. Glasgow Mathematical Journal, (3): Bacon, M. R. and Kappe, L. C. On Capable p-groups of Nilpotency Class Two. Illinois Journal of Mathematics, : Eick, B. and Nickel, W. Computing Schur Multiplicator and Tensor Square of Polycyclic Group. Journal of Algebra, (2): Eick, B. and Nickel. W. Polycyclic- Computing with Polycyclic Groups, A GAP package. 27. Rotman, J.J. An Introduction to the Theory of Groups. 4th. ed. New York: Springer-Verlag Hungerford, T.W. Graduate Texts in Mathematics: Algebra. New York: Springer-Verlag Brown, K. S. Cohomology of Groups. New York: Springer-Verlag Segal, D. Polycyclic Groups. Cambridge: Cambridge University Press Blyth, R. D., Fumagalli, F. and Morigi, M. Some Structural Results on The Nonabelian Tensor Square of groups. Journal of Group Theory, (1): Blyth, R. D. and Moravec, P. and Morse, R. F. On the Nonabelian Tensor Squares of Free Nilpotent Groups of Finite Rank. Contemporary Mathematics, : Magidin, A. and Morse, R.F. Certain Homological Functors. Contemporary Mathematics, : Higman, G. Finite Groups Having Isomorphic Images In Every Finite Group of Which They Are Homomorphic Images. Quart. Journal of Math. Oxford, :